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Robust design in multibody dynamics - application to vehicle ride-comfort optimization
Anas Batou, Christian Soize, C. K. Choi, H. H. Yoo
To cite this version:
Anas Batou, Christian Soize, C. K. Choi, H. H. Yoo. Robust design in multibody dynamics - application to vehicle ride-comfort optimization. IUTAM Symposium on Dynamical Analysis of Multibody Systems with Design Uncertainties, IUTAM, Jun 2014, Stuttgart, Germany. pp.90-97,
�10.1016/j.piutam.2015.01.005�. �hal-01011920�
IUTAM Symposium on “Dynamical Analysis of Multibody Systems with Design Uncertainties”
Robust design in multibody dynamics - application to vehicle ride-comfort optimization
Anas Batoua,∗, Christian Soizea, Chan Kyu Choib, Hong Hee Yoob
aUniversit´e Paris-Est, Mod´elisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 Bd Descartes, 77454 Marne-la-Vall´ee, France
bSchool of Mechanical Engineering, Hanyang University,17 Haengdang-Dong Sungdong-Gu, Seoul 133-791, South Korea
Abstract
This research is devoted to the robust design of multibody dynamical systems, that is to say to the optimal design of a multibody system which is carried out using an uncertain computational model. The probabilistic model of uncertainties is constructed using a probabilistic approach yielding a stochastic differential equation with random initial conditions. Then the robust design of the multibody system, in presence of uncertainties, is performed using the least-square method for optimizing the cost function, and the Monte Carlo simulation method as stochastic solver.
The application consists in a simple multibody model of an automotive vehicle crossing a rough road and for which the suspensions have to be designed in order to optimize the comfort of the passengers.
Keywords: Type your keywords here, separated by semicolons ;
1. Introduction
This paper deals with the robust design of a multibody dynamical system using a computational model for which some parameters are uncertain. The uncertainties are induced by natural variability or by a lack of knowledge existing on these parameters. For a multibody system, uncertainties can affect (1) the bodies themselves (inertia properties), (2) the joints between the bodies, (3) the external forces. If they are not negligible, theses uncertainties have to be taken into account in order to predict the response of the uncertain multibody system with a good robustness. In the context of the design of a multibody system, these uncertainties are propagated into the performance function that has to be optimized. Then the robust design consists in searching the optimal design of the multibody system taking into account uncertainties, and then allows the robustness of this optimal design o be analyzed with respect to uncertainties.
In the context of multibody dynamics, the parametric probabilistic approach of uncertainties consists in model- ing the uncertain parameters of the multibody dynamical systems by random variables1,4,10,12,15,18,19,22. Therefore, the quantities of interest and the performance function become random variables, and a probabilistic robust design method3,5,7,9,11,13,14,17,25has to be used.
The first objective of this paper consists in constructing a probabilistic model of uncertainties yielding a stochastic differential equation with random initial conditions. The second one is related to the robust design of a multibody system in presence of uncertainties by using the least-square method for constructing the cost function, and the Monte Carlo simulation method as stochastic solver.
∗Corresponding author. Tel.:+33-160-957-792 ; fax:+33-160-957-799.
E-mail address:anas.batou@univ-paris-est.fr
In Section 2, the nominal model is presented for the rigid multibody system. Then, the stochastic model of uncer- tainties and the robust design method are presented in Sections 3 and 4. Finally, Section 4 presents an application that consists in a simple multibody model of an automotive vehicle crossing a rough road, and for which the suspensions have to be designed in order to optimize the comfort of the passengers9.
2. Nominal model for the rigid multibody dynamical system
The mean model is constructed as follows21,20. Let RBibe one of the rigid bodies, occupying a bounded domain Ωiwith a given geometry. Each rigid body RBi is represented by its massmi, the position vectorriof its center of mass, and by the matrix [Ji] of its tensor of inertia defined in the local frame. The multibody dynamical system is made up ofnbrigid bodies and of some ideal joints. The interactions between the rigid bodies are realized by these ideal joints, but also by springs, dampers, and actuators, which produce forces between the bodies. Letube the vector inR6nb such thatu =(r1, ...,rnb,s1, ...,snb) in whichsi =(αi, βi, γi) is the rotation vector. Thencconstraints induced by the joints are given byncimplicit equations which are globally written asϕ(u,t)=0. Then the function {u(t),∈[0,T]}is the solution of the following differential equation
[ [M] [ϕu]T [ϕu]
[0]
] [ u¨ λ
]
=
[ q−k
−dtdϕt−[dtdϕu] ˙u ]
, (1)
with the initial conditions
u(0)=u0 , u(0)˙ =v0 , (2)
in which [M] is the (6nb×6nb) mass matrix,k( ˙u) is the vector of the Coriolis forces, [ϕu(u(t),t)]i j=∂ϕi(u(t),t)/∂uj(t), andϕt =∂ϕ/∂t. The vectorq(u,u,˙ t) is constituted of the applied forces and torques induced by springs, dampers, and actuators. The vectorλ(t) is the vector of the Lagrange multipliers.
The performance of the multibody system is measured by a performance function,g(u), with values inRng. 3. Stochastic model of uncertainties and robust design
For a multibody system, the possible sources of parametric uncertainties are the following.
(i)- Uncertainties for the spatial mass distribution inside a body. For instance, such a type of uncertainties can be encountered for a vehicle in which the passengers have a mass and a position that are variable. For each body, this type of uncertainties yields a random mass, a random position of the center of mass, and a random tensor of inertia.
Consequently, the mass matrix [M] and the Coriolis forcesKare random. The probability density functions (pdf) of these random masses, random positions of the centers of mass, and random tensors of inertia have been constructed1 using the Maximum Entropy Principle, in which a special care has been devoted to the probabilistic modeling of the random tensor of inertia in following the methodology of the nonparametric probabilistic approach of uncertainties introduced in23,24for the construction of random matrices.
(ii)- Uncertainties in the joints. For the ideal joints, the directions and the points defining the joints can be uncertain.
These uncertainties may be due to manufacturing tolerances, or due to the natural wear during the life cycle of the multibody system. Such uncertainties have to be taken into account in order to ensure a good accuracy for the pre- diction of the dynamical response of the multibody system. For non-ideal joints the friction coefficients can also be uncertain. The randomness in the joints between the bodies yields a random constraint vectorΦin Eq. (1).
(iii)- Uncertainties in internal forces. Concerning internal forces, there may be uncertainties in the constitutive laws of the multidimensional springs and dampers. In such a case, uncertainties may be taken into account in the param- eters of the constitutive laws, or directly in the stiffness and damping matrices using the nonparametric probabilistic approach of uncertainties (see23). The uncertainties in the internal forces induce a random vectorQin Eq. (1).
2
Letxbe the vector of thenuncertain parameters of the multibody system. Since the uncertainties in the system parameters are taken into account using a probabilistic approach, thenxis modeled by a random vectorXwith values inRn. Random vectorXis written asX=(Xd,Xf) in whichXdis the random vector with values inRndof the random design parameters, andXf is the random vector with values inRnf of the fixed (but random) system parameters (we then haven =nd+nf). Apriorprobabilistic model of random vectorXmust be constructed and/or identified using an adapted methodology24. Thispriorprobabilistic model depends on hyperparameters which are the mean values and other quantities allowing the statistical fluctuations (level of uncertainties) and the statistical dependencies between parameters to be controlled (coefficients of variation, correlation matrices, etc). In the context of uncertainty quantification, two cases can be considered. If experimental data are available, then the hyperparameters of theprior probabilistic model can be identified solving an inverse statistical problem. If there are no experimental data, then the mean values are chosen as the nominal values and the other hyperparameters can be considered as sensitivity parameters in order to analyze the robustness of the responses with respect to the level of uncertainties. In the context of robust design, the design parameters are chosen as the mean values of thepriorprobabilistic model of random design parameters,Xd, while the other hyperparameters of thepriorprobabilistic model ofXdare treated as previously explained. In this paper, since no experimental data are available, the design parameters will be the mean values ofXd and all the other hyperparameters of thepriorprobabilistic model ofXwill be either fixed or used as sensitivity parameters. In this last case, the robustness of the optimal design point can be analyzed with respect to the level of uncertainties.
LetU=(R1, ...,Rnb,S1, ...,Snb) be theR6nb-valued stochastic processes indexed by [0,T], which model the 6nb
random coordinates. LetΛbe the Rnc-valued stochastic process indexed by [0,T], which models thencrandom Lagrange multipliers. The deterministic Eq. (1) becomes the following stochastic equation
[M] [ ϕu]T
[ϕu] [0]
[ U¨
Λ ]
=
[ Q−K
−dtdϕt−[d
dtϕu] U˙
]
, (3)
U(0)=U0 , U(0)˙ =v0, (4)
in whichU0is a given random vector, andv0is a given deterministic vector. Random Eqs. (3) and (4) are solved using the Monte Carlo simulation method.
The performance of the stochastic multibody system is measured by theRng-random variable,G=g(U), which is assumed to be a second-order random variable.
4. Robust design
As explained in the previous section, the vector-valued design parameter is the vectord =E{Xd}with values in Rnd, which is the mean vector of the random design parameterXdwith values inRnd, and whereEis the mathematical expectation. All the other hyperparameters of the probabilistic model of random vectorX =(Xd,Xf) are assumed to be fixed (nevertheless, these other hyperparameters can be used for carrying out a sensitivity analysis of the op- timal design point with respect to the level of uncertainties). LetCd ⊆Rnd be the admissible set for vectord. The performance random vector,G, depends ond, and is then rewritten asGd =g(Ud). We obtain a family of random variables{Gd,d ∈ Cd}. LetGd = E{Gd}be the mean value ofGd. Letg∗be the deterministic target performance vector associated with performance random vectorGd. The optimal valuedoptof vectordis calculated using the least square method,i.e,
dopt =arg min
d∈CdD(d), (5)
in which the cost functionD(d) is written as
D(d)=E{∥Gd−g∗∥2}=E{∥Gd−Gd∥2}+∥g∗−Gd∥2, (6) which means that the minimization problem defined by Eq. (5) aims at minimizing both (i) the bias between the mean value of the random performance vector and the deterministic target performance vector and (2) the variance of the
random performance vector.
For each value of vectord, functionD(d) is estimated using the Monte Carlo simulation method.
5. Application
The application is devoted to the stochastic response of a vehicle (see its scheme in Fig. 1) at constant speed excited by the ground elevation that is assumed to be a homogeneous random field. Consequently, the imposed vertical displacement is a stationary stochastic process. The computational model of the multibody system is uncertain, and its input is a stationary stochastic excitation.
5.1. Definition of the nominal multibody system
The system is made up of 9 rigid bodies: The sprung mass RB1(the vehicle body), the two front unsprung masses RB2fr, RB3fr (front wheels), the two rear unsprung masses RB2re, RB3re (rear wheels), the two front massless bodies RB4fr, RB5fr, the two rear massless bodies RB4re, RB5re, the two front seats (with passengers) RB6fr, RB7fr, and the two rear seats (with passengers) RB6re, RB7re. The sprung mass RB3is linked to each seat by four springs and four dampers.
The sprung mass RB3is linked to each unsprung mass by a spring and a damper. Each unsprung mass is linked to a massless body by a spring and a damper. Each massless body is linked to the ground by a prismatic joint following the vertical direction. LetLbe the distance between the two axles and letV be the speed of the car. The responses are analyzed fort in [0,T], in whichT = 100 s is the final time. The front imposed displacements are modeled by independent stationary Gaussian stochastic processes denoted by{t7→ Z1fr(t),t ≥0}and{t 7→ Z2fr(t),t ≥0}. The corresponding rear imposed displacements stochastic processes are{t7→Z1re(t),t≥0}and{t7→Z2re(t),t≥0}such that (1) fort∈[0,L/V[,Z1re(t)=0 andZ2re(t)=0 and (2) fort∈[L/V,T],Z1re(t)=Z1fr(t−L/V) andZ2re(t)=Z2fr(t−L/V).
The power spectral density (PSD) functions of stochastic processesZ1fr(t) and Z2fr(t) are equal, and are plotted in
Figure 1: Nominal multibody system: Schematic front view (left figure), 3D view (right figure).
Fig. 2. This PSD corresponds to a D-class road roughness in the ISO classification8,9. Figure 3 shows a realization of stochastic processZ1fr(t). LetP1be an observation point belonging to the front-left seat andP2be an observation point belonging to the rear-left seat. The PSD function of the random transient vertical acceleration at pointP1and at pointP2are estimated using the periodogram method16, and are plotted in Fig. 4.
4
10−2 10−1 100 101 102 10−8
10−7 10−6 10−5 10−4 10−3 10−2
Frequency (Hz) PowSpec (m2 Hz−1 )
Figure 2: Power spectral density function of stochastic processesZ1frandZ2fr.
0 20 40 60 80 100
−0.2
−0.15
−0.1
−0.05 0 0.05 0.1 0.15
Time (s)
Displacement (m)
0 0.1 0.2 0.3 0.4 0.5
−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06
Time (s)
Displacement (m)
Figure 3: Realization of stochastic processZ1fr(left figure), and its zoom in [0,5] s (right figure).
5.2. Probability model of fixed system-parameter uncertainties
In this application, only the fixed system parameters are uncertain, and the design parameters are assumed to be deterministic. The uncertain fixed parameters of the multibody system are the inertia properties of the sprung mass, the inertia properties of the seats, the stiffnesses of the unsprung-masses/massless springs, the damping coefficient of the unsprung-masses/massless dampers. The random inertia properties (random mass, random center of mass, and random tensor of inertia) are constructed as follows1. The random stiffnesses and the random damping coefficients are Gamma random variables with coefficient of variation (ratio between the standard deviation and the mean value) equal to 0.1.
5.3. Robust design
The design parameters are (1) the damping coefficients of the dampers between the sprung mass and the unsprung masses, which are such thatd3fr=d4fr=d3re=d4re, and (2) the damping coefficients of the dampers between the sprung mass and the seats, which are such thatd5fr =d6fr = d5re = d6re. Therefore, there arend =2 design parameters and d=(d3fr,d5fr). Its initial value isdinit=(2217,318) N s m−1.
For a given realization of the random vectorXf (modeling the uncertainties for the fixed system parameters), the performance function is related to the ride comfort index8,9, which is expressed as a function of the PSD functions of the vertical accelerations at given points (which are stationary stochastic processes due to the stationarity of the stochastic input). Considering all the possible realizations (following the probability model) of the random vectorXf,
0 10 20 30 40 50 10−5
10−4 10−3 10−2 10−1
Frequency (Hz) PowSpec (m2 s−4 Hz−1 )
0 10 20 30 40 50
10−4 10−3 10−2 10−1
Frequency (Hz) PowSpec (m2 s−4 Hz−1 )
Figure 4: Power spectral density function of the stationary stochastic acceleration at pointP1(left figure) and at pointP2(right figure).
the ride comfort index,Gd=(G1,d,G2,d), becomes a random vector such that
G1,d=
√
∫ f2
f1
w(f)2S1d(2πf)d f,
G2,d=
√
∫ f2
f1
w(f)2S2d(2πf)d f,
(7)
in whichf 7→S1d(2πf) and f 7→Sd2(2πf) are the random PSD functions of the stationary stochastic vertical accelera- tion at pointP1and at pointP2, and where f 7→w(f) is the weighting function plotted in Fig. 5.
100 101
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
Frequency (Hz)
Weight
Figure 5: Weighting functionf 7→w(f).
The cost functiond 7→ D(d) is plotted in Fig. 6. In this figure, it can be seen that the optimal value isdopt = (665,669) N s m−1. The corresponding probability density functions of the random ride comfort indices,G1,dopt and G2,dopt, are plotted in Fig. 7. The mean values and the variances ofG1,dopt and ofG2,doptare reported in Table 1. As expected, it can be seen in Fig. 7 and in Table 1 that, compared to the initial design configuration, the mean values and the variances of the random ride comfort indices are lower for the optimal design configuration.
6
500 1000 1500 200
300 400 500 600 700 800
d5 fr
d 3fr
0.13 0.135 0.14
Figure 6: Cost functiond7→ D(d).
0 0.5 1 1.5
0 1 2 3 4 5
0 0.5 1 1.5 2
0 1 2 3 4 5
Figure 7: Solid lines: Probability density functions ofG1,dopt(left figure) andG2,dopt(right figure). Dashed lines: Probability density functions of G1,dinit(left figure) andG2,dinit(right figure).
Initial configuration Optimal configuration Mean value ofG1,d 0.82 m s−2 0.68 m s−2 Mean value ofG2,d 0.90 m s−2 0.69 m s−2 Variance ofG1,d 1.2×10−2m2s−4 7.8×10−3m2s−4 Variance ofG2,d 1.7×10−2m2s−4 9.1×10−3m2s−4
Table 1: Statistics forG1,doptandG2,dopt.
6. Conclusions
A methodology has been presented for analyzing the robust design of multibody systems in presence of uncertain- ties. The uncertainties have been taken into account using a parametric probabilistic approach. The optimal design parameters have been calculated using the least-square method. The methodology has been validated on a multi- body system representing an automotive vehicle crossing a rough road. The optimal damping coefficients have been calculated in order to optimize the ride comfort index.
Acknowledgements
This research was supported by the ”National Research Foundation of Korea” (NRF, Grant numbers: NRF- 2013K1A3A1A21000356) and the ”French Ministry of Foreign Affairs” (Grant numbers: 29892TM), through the STAR program.
References
1. Batou A, Soize C. Rigid multibody system dynamics with uncertain rigid bodies.Multibody System Dynamics2012;27(3):285-319.
2. Bestle D, Eberhard P. Multicriteria multimodel design optimization.IUTAM Symposium on Optimization of Mechanical Systems:33-40, Kluwer Academic Publishers, Dordrecht; 1996.
3. Capiez-Lernout E, Soize C. Robust design optimization in computational mechanics.Journal of Applied Mechanics - Transactions of the ASME 2008;75(2):1-11.
4. Carrarini A. Reliability based analysis of the crosswind stability of railway vehicles.Journal of Wind Engineering and Industrial Aerodynamics 2007;95:493-509.
5. Choi C K, Yoo H H. Uncertainty analysis of nonlinear systems employing the first-order reliability method.Journal of Mechanical Science and Technology2012;26(1):39-44.
6. Datoussa¨ıd S, Hadjit R, Verlinden O, Conti C. Optimization design of multibody systems by using genetic algorithms 1998.Vehicle System Dynamics;29(1):704-710.
7. Han H, Park T. Robust optimal design of multi-body systems.Multibody System Dynamics2004;11(2):167-183.
8. International Organization for Standardization ISO/TC108/SC2/WG4 N57.Reporting vehicle road surface irregularities; 1982.
9. Kim B S, Yoo H H. Ride comfort uncertainty analysis and reliability design of a passenger vehicle undergoing random road excitation.Proc IMechE Part D: J Automobile Engineering2013;227(3):433-442.
10. Li L, Corina Sandu C. On the impact of cargo weight, vehicle parameters, and terrain characteristics on the prediction of traction for off-road vehicles.Journal of Terramechanics2007;44:221-238.
11. Mbaye M, Soize C, Ousty J-P, Capiez-Lernout E. Robust analysis of design in vibration of turbomachines.ASME Journal of Turbomachinery 2013;135(2):021008-1–8,.
12. Negrut D, Datar M, Gorsich D, Lamb D. A framework for uncertainty quantification in nonlinear multi-body system dynamics.Proceedings of the 26th Army Science Conference2008; Orlando, FL.
13. Papadrakakis M, Lagaros N D. Reliability-based structural optimization using neural networks and Monte Carlo simulation.Computer Methods in Applied Mechanics and Engineering2002;191(32):3491-3507.
14. Park G J, Lee T H, Lee K H, Hwang K H. Robust design: an overview.AIAA Journal2006;44(1):181-191.
15. Perrin G, Soize C, Duhamel D, Funfschilling C, Track irregularities stochastic modeling.Probabilistic Engineering Mechanics2013;34:
123-130.
16. Priestley M B.Spectral Analysis and Time Series. Academic Press, New York; 1981.
17. Rahmani Hanzaki A, Rao P V M, Saha S K. Kinematic and sensitivity analysis and optimization of planar rack-and-pinion steering linkages.
Mechanism and Machine Theory2009;44(1):42-56.
18. Sandu A, Sandu C, Ahmadian M. Modeling multibody dynamic systems with uncertainties. Part I: theoretical and computational aspects.
Multibody System Dynamics2006;23:375-395.
19. Sandu A, Sandu C, Ahmadian M. Modeling multibody dynamic systems with uncertainties. Part II: numerical applications.Multibody System Dynamics2006;22:241-262.
20. Shabana A A.Dynamics of Multibody Systems. Cambridge University Press; 2013.
21. Schiehlen W.Multibody Systems Handboob. Springer-Verlag, Berlin; 1990.
22. Schmitt K P, Anitescu M , Negrut D. Efficient sampling for spatial uncertainty quantification in multibody system dynamics applications.Int.
J. Numer. Meth. Engng.2009;80:537-564.
23. Soize C. A nonparametric model of random uncertainties for reduced matrix models in structural dynamics.Probabilistic Engineering Me- chanics2000;15(3):277-294.
24. Soize C.Stochastic Models of Uncertainties in Computational Mechanics. American Society of Civil Engineers (ASCE), Reston; 2012.
25. Sundaresan S, Ishii K, Houser D R. A robust optimization procedure with variations on design variables and constraints.Engineering Opti- mization1995;24(2):101-117.
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